Let $M = (m_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$ such that for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. We assume in this question that there exist $U_1, U_2, \cdots, U_n$ elements of $\mathbb{R}^p$ such that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} = \|U_i - U_j\|^2$. 1) Show that the eigenvalues of $\Phi(M)$ are all real and non-positive. 2) We further assume (if necessary by performing a translation) that the $(U_i)_{i \in \llbracket 1,n\rrbracket}$ are centered, that is, $\sum_{i=1}^n U_i = 0$. Show that $\operatorname{rg}(U) = \operatorname{rg}(U_1 | U_2 | \cdots | U_n) = \operatorname{rg}(\Phi(M))$ and that $p \geqslant \operatorname{rg}(\Phi(M))$.
Let $M = (m_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$ such that for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. We assume in this question that there exist $U_1, U_2, \cdots, U_n$ elements of $\mathbb{R}^p$ such that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} = \|U_i - U_j\|^2$.
1) Show that the eigenvalues of $\Phi(M)$ are all real and non-positive.
2) We further assume (if necessary by performing a translation) that the $(U_i)_{i \in \llbracket 1,n\rrbracket}$ are centered, that is, $\sum_{i=1}^n U_i = 0$.
Show that $\operatorname{rg}(U) = \operatorname{rg}(U_1 | U_2 | \cdots | U_n) = \operatorname{rg}(\Phi(M))$ and that $p \geqslant \operatorname{rg}(\Phi(M))$.